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%I #18 Apr 12 2023 11:06:25
%S 1,1,1,1,2,5,11,21,38,71,141,289,591,1195,2410,4897,10051,20763,42996,
%T 89139,185170,385809,806349,1689573,3547152,7459715,15714655,33161821,
%U 70095642,148388521,314562189,667682057,1418942341
%N Generalized (3,-1) Catalan numbers.
%C Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(3,-1). Hankel transform has g.f. (1-x^3)/(1+x^4) (A132380 (n+3)).
%H G. C. Greubel, <a href="/A144700/b144700.txt">Table of n, a(n) for n = 0..1000</a>
%H S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017).
%F G.f.: (1/(1-x)) * c(x^4/(1-x)^3), where c(x) is the g.f. of A000108.
%F a(n) = Sum_{k=0..floor(n/4)} binomial(n-k, 3*k)*A000108(k).
%F (n+4)*a(n) = 2*(2*n+5)*a(n-1) - 6*(n+1)*a(n-2) + 2*(2*n-1)*a(n-3) +3*(n-2)*a(n-4) - 2*(2*n-7)*a(n-5). - _R. J. Mathar_, Nov 16 2011
%F a(n) = b(n, 3), where b(n, m) = Sum_{k=0..floor(n/(m+1))} binomial(n-k, m*k)*A000108(k). - _G. C. Greubel_, Jun 15 2022
%t b[n_, m_]:=a[n, m]=Sum[Binomial[n-k,m*k]*CatalanNumber[k], {k,0,Floor[n/(m+1)]}];
%t A144700[n_]:= b[n,3]; (* A014137 (m=0), A090344 (m=1), A023431 (m=2) *)
%t Table[A144700[n], {n, 0, 40}] (* _G. C. Greubel_, Jun 15 2022 *)
%o (Magma) [(&+[Binomial(n-k,3*k)*Catalan(k): k in [0..Floor(n/4)]]): n in [0..40]]; // _G. C. Greubel_, Jun 15 2022
%o (SageMath) [sum(binomial(n-k,3*k)*catalan_number(k) for k in (0..(n//4))) for n in (0..40)] # _G. C. Greubel_, Jun 15 2022
%Y Cf. A000108, A014137, A023431, A090344, A132380.
%K easy,nonn
%O 0,5
%A _Paul Barry_, Sep 19 2008
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